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In this article we study a general class of goodness-of-fit tests for the conditional mean of a linear or nonlinear time series model. Among the properties of the proposed tests are that they are suitable when the conditioning set is infinite-dimensional; are consistent against a broad class of alternatives including Pitman's local alternatives converging at the parametric rate; and do not need to choose a lag order depending on the sample size or to smooth the data. It turns out that the asymptotic null distributions of the tests depend on the data generating process, so a new bootstrap procedure is proposed and theoretically justified. The proposed bootstrap tests are robust to higher order dependence, in particular to conditional heteroskedasticity of unknown form. A simulation study compares the finite sample performance of the proposed and competing tests and shows that our tests can play a valuable role in time series modeling. Finally, an application to an economic price series highlights the merits of our approach.

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